NUMERICAL INVESTIGATION OF THE PERFORMANCE OF INTERIOR PERMANENT

MAGNET SYNCHRONOUS MOTOR DRIVE

SHAHIDA PERVIN

THESIS SUBMITTED IN FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE IN MATHEMATICS

FACULTY OF SCIENCEUNIVERSITY OF MALAYA

KUALA LUMPUR

2014

UNIVERSITI MALAYA

ORIGINAL LITERARY WORK DECLARATION

Name of Candidate: Shahida Pervin (I.C/Passport No: WS587463)

Registration/Matric No: SGP 110002

Name of Degree: Master of Science

Title of Project Paper/Research Report/Dissertation/Thesis (“this Work”): Numerical Analysis Based Performance Investigation of Interior Permanent Magnet Synchronous Motor Drive

Field of Study: Numerical Analysis of Motor Drives

I do solemnly and sincerely declare that:

(1) I am the sole author/writer of this Work; (2) This Work is original; (3) Any use of any work in which copyright exists was done by way of fair dealing and for

permitted purposes and any excerpt or extract from, or reference to or reproduction of any copyright work has been disclosed expressly and sufficiently and the title of the Work and its authorship have been acknowledged in this Work;

(4) I do not have any actual knowledge nor ought I reasonably to know that the making of this work constitutes an infringement of any copyright work;

(5) I hereby assign all and every rights in the copyright to this Work to the University of Malaya (“UM”), who henceforth shall be owner of the copyright in this Work and that any reproduction or use in any form or by any means whatsoever is prohibited without the written consent of UM having been first had and obtained;

(6) I am fully aware that if in the course of making this Work I have infringed any copyright whether intentionally or otherwise, I may be subject to legal action or any other action as may be determined by UM.

Candidate’s Signature Date

Subscribed and solemnly declared before,

Witness’s Signature Date

Name:

Designation:

ABSTRACT

The interior permanent magnet synchronous motor (IPMSM) is getting popular in

industrial drives because of its' advantageous features such as high power to weight

ratio, high efficiency, high power factor, low noise, and robustness. The vector control

technique is widely used for high performance IPMSM drive as the motor torque and

flux can be controlled separately. Fast and accurate response, quick recovery of speed

from any disturbances and insensitivity to parameter variations are some of the

important characteristics of high performance drive system used in electric vehicles,

robotics, rolling mills, traction and spindle drives.

Despite many advantageous features of IPMSM, precise control of this motor at

high-speed conditions especially, above the rated speed remains an engineering

challenge. At high speeds the voltage, current and power capabilities of the motor

exceed the rated limits. Consequently, the nonlinearity due to magnetic saturation of the

rotor core and hence the variation of d- and q- axes flux linkages will be significant.

Thus, it severely affects the performance of the drive at high speeds. The IPMSM drive

can be operated above the rated speed using the field-weakening (FW) technique. In

IPMSM the flux/field can only be weakened by the demagnetizing effect of d-axis

armature reaction current, id. Recently, researchers developed FW control algorithms

but often ignored the high precision computation of the algorithm. Mostly they simplify

the equations of flux control by ignoring the stator resistance and depend on

MATLAB/Simulink library. This results in improper flux weakening operation of

IPMSM. However, the proper flux computation is a crucial issue for motor control

particularly, at high speed condition. Therefore, there is a need to investigate the other

computational methods.

iii

In this study, accurate flux estimation for proper FW operation of IPMSM is

developed by incorporating the stator resistance of the motor. The Newton-Raphson

method (NRM) based numerical computation is used for high precision computation of

flux component of stator current, id to enhance the performance of the IPMSM drive

over wide speed range. The performance of the proposed NRM based computation of id

for IPMSM drive is evaluated in simulation using MATLAB/Simulink at different

operating conditions. The performance of the IPMSM drive with the proposed NRM

method is also compared with the conventional simplified computation of id. It has been

found that the IPMSM drive with proposed calculation of id provides better response as

compared to the conventional calculation of id. Thus, the proposed method could be a

potential candidate for real-time field weakening operation of IPM motor.

In the next step, fourth order Runge-Kutta method (RKM) is used to solve the motor

differential equations and the performance of the IPMSM drive is tested and compared

with the MATLAB/Simulink library built-in motor model. The results found that there

is no significant difference between RKM based calculation and MATLAB/Simulink

library. Thus, the MATLAB/Simulink library motor model can be used for motor drives

simulation with sufficient accuracy.

iv

ABSTRAK

Pemacu motor segerak magnet dalaman kekal (PMSMDK) semakin popular dalam

industri pemacu kerana kelebihan cirinya seperti berkuasa tinggi berbanding nisbah,

berkecekapan tinggi, faktor kuasa tinggi, bunyi rendah dan kekukuhan. Teknik kawalan

vektor digunakan secara meluas untuk pemacu PMSMDK berprestasi tinggi kerana tork

motor dan fluks boleh dikawal secara berasingan. Tindak balas cepat dan tepat, kelajuan

dipulihkan segera dari sebarang gangguan dan tidak sensitif kepada variasi parameter

adalah beberapa ciri penting sistem pemacu berprestasi tinggi yang digunakan dalam

kenderaan elektrik, robotik, mesin penggelek, peranti tarikan dan pengumpal.

Walaupun banyak ciri berfaedah PMSMDK, kawalan persis motor ini pada kelajuan

tinggi terutamanya di atas laju kadaran kekal sebagai cabaran kejuruteraan. Pada

kelajuan tinggi, voltan, arus dan keupayaan tenaga motor melebihi had laju kadaran.

Oleh itu, , ketaklinearan disebabkan penepuan magnet teras rotor dan dengan itu variasi

hubungan paksi-d dan -q flux menjadi penting. Oleh itu, ianya memberi kesan kepada

prestasi pemacu pada kelajuan tinggi. Pemacu PMSMDK boleh beroperasi di atas laju

kadaran menggunakan teknik medan lemah (ML). Dalam IPMSM fluks/medan hanya

menjadi lemah oleh kesan penyahmagnetan paksi-d arus tindakbalas armatur, id.

Baru-baru ini, penyelidik membangunkan algoritma kawalan ML tetapi sering

mengabaikan algoritma pengiraan kejituan tinggi. Kebanyakan mereka meringkaskan

persamaan kawalan fluks dengan mengabaikan rintangan pemegun dan bergantung

kepada perpustakaan MATLAB/Simulink. Keputusan ini menyebabkan operasi

PMSMDK medan lemah tidak wajar. Walau bagaimanapun, pengiraan fluks yang betul

adalah isu penting untuk kawalan motor, terutamanya pada keadaan kelajuan tinggi.

Oleh itu, ianya perlu untuk mengkaji kaedah pengiraan lain.

v

Dalam kajian ini, anggaran fluks tepat untuk operasi ML PMSMDK dibangunkan

dengan menggabungkan motor rintangan pemegun Kaedah Newton-Raphson (KNR)

berasaskan pengiraan berangka digunakan untuk membuat pengiraan ketepatan tinggi

komponen fluks arus pemegun, id untuk meningkatkan prestasi pemacu PMSMDK

melangkaui julat kelajuan. Prestasi pengiraan KNR berdasarkan cadangan id PMSMDK

dinilai dalam simulasi menggunakan MATLAB/Simulink pada keadaan operasi yang

berbeza. Prestasi pemacu PMSMDK dengan KNR yang dicadangkan juga berbanding

dengan pengiraan id konvensional yang dipermudahkan. Didapati pemacu PMSMDK

dengan pengiraan id yang dicadangkan member respon lebih baik berbanding dengan

pengiraan id konvensional. Oleh itu, kaedah yang dicadangkan itu boleh menjadi bahan

berpotensi untuk masa sebenar operasi motor MDK medan lemah.

Seterusnya, kaedah Runge-Kutta (KRK) peringkat empat digunakan untuk

menyelesaikan persamaan pembezaan motor dan prestasi pemacu PMSMDK diuji dan

dibandingkan dengan perpustakaan model motor MATLAB/Simulink terbina dalaman.

Keputusan didapati bahawa terdapat perbezaan yang signifikan antara pengiraan

berdasarkan KRK dan perpustakaan MATLAB/Simulink. Oleh itu, perpustakaan model

motor MATLAB/Simulink boleh digunakan untuk simulasi pemacu motor dengan

ketepatan yang mencukupi.

vi

Acknowledgements

I would like to express my most sincere gratitude and appreciation to my supervisor

Dr. Zailan bin Siri for his guidance, advice and encouragement throughout of this study.

I wish to thank Prof. Dr. Angelina Chin Yan Mui (Postgraduate Coordinator, Institute of

Mathematical Sciences) for her help although out my study at University of Malaya. I

also offer my sincere thanks to Professor Dr. M. Nasir Uddin who provides me all the

support on the technical side of motor drives.

I would like to acknowledge the assistance from the Institute of Mathematical

Sciences, University of Malaya, all Faculty members, graduate fellows and staff

members.

Finally, I express my sincere appreciation to my husband, daughter, as well as other

family members, relatives and friends without whose support and encouragement it

would not have been possible to complete this study.

vii

DEDICATED TO:

My husband Prof. Dr. M. Nasir Uddin

and,

My daughter Miss Shirley Naima Uddin

viii

TABLE OF CONTENTS

Page

ABSTRACT iii

ABSTRAK MALAYA v

ACKNOWLEDGEMENTS vii

LIST OF FIGURES xiii

LIST OF SYMBOLS xv

LIST OF ACRONYMS xvii

1 Introduction 1

1.1 Background of Motor

1.1.1 Direct Current (DC) Motor

1.1.2 Alternating Current (AC) Motor

1.2 Background of the Permanent Magnet Synchronous Motor (PMSM)

ix

1.2.1 Classifications of PMSM

1.3 Motivation and Objectives

1.4 Methodology

1.5 Introduction to Numerical Analysis Methods

1.5.1 Newton-Raphson Method

1.5.2 Runge-Kutta Method

1.6 Organizations of the Thesis

2 Literature Review 15

2.1 Mathematical Model of Interior Permanent Magnet Synchronous Motor

(IPMSM)

2.2 Vector Control Strategy for IPMSM Drives

2.2.1 Speed Controller

2.2.2 Current Controller and Voltage Source Inverter

2.3 Flux Control Methods for IPMSM Drives

2.4 Application of Numerical Analysis for Motor Drives

2.5 Conclusion

x

3 Flux Control Techniques for IPMSM Drive 26

3.1 Introduction

3.2 Conventional Flux Control of IPMSM

3.2.1 Conventional Flux Control below Rated Speed – Maximum Torque

per Ampere (MTPA) Control

3.2.2 Conventional Flux Control above Rated Speed – Field Weakening

(FW) Control

3.3 Proposed Flux Control Techniques

3.3.1 Proposed Flux Control below Rated Speed

3.3.2 Proposed Flux Control above Rated Speed

3.4 Simulation Results and Discussions

3.5 Experimental Test of the Proposed Field Weakening

3.6 Conclusion

4 Application of Runge-Kutta Method for Solution of IPMSM Model 46

4.1 Introduction

4.2 Modelling of IPMSM

4.3 Simulation Results of the RKM and MATLAB/Simulink Based Motor

Model

4.4 Conclusion

xi

5 Conclusion 51

5.1 Summary of the Thesis

5.2 Major Achievements of the Current Work

5.3 Further Scope of the Work

List of Papers Published from this Work 55

REFERENCES 56

APPENDICES 61

xii

List of Figures

1.1 Structure of a simple 2-pole PM dc motor ................................................ 3

1.2 Squirrel-cage induction motor. .............................................................. 4

1.3 Cross section of a 4-pole: (a) surface mounted, (b) inset and (c) IPM type motor.. 7

2.1 Equivalent circuit model of the IPMSM: (a) d-axis, (b) q-axis.…………. 18

2.2 Basic vector diagram of IPMSM: (a) general; (b) modified with id=0………… 20

2.3 Block diagram of the closed loop vector control of IPMSM drive……....….. 21

3.1 Typical torque-speed characteristic curve of a motor drive……………….... 32

3.2 Locus of stator current Ia at different speeds staring from 188 rad/s to 363 rad/s in a

step of 25 rad/s. ………………………………………………………........... 32

Fig. 3.3: Variation of id with speed, r…………………………………. 36

Fig. 3.4: Comparison of curve fitting polynomial with actual id. . ……… 36

Fig. 3.5. (a) Simulink block diagram for overall control system of IPMSM drive, (b)

Reference current generator subsystem. ………………………………. 38

Fig. 3.6: Simulated transient responses of the drive for step change of speed at rated

load using the conventional computation of id [12]; (a) speed, (b) iq*, (c) id

*, (d) stator

phase current, ia(actual and command)……… ……………………….……….... 39

3.7: Simulated transient responses of the IPMSM drive for step change of speed at rated

load using the proposed NRM based id computation; ; (a) speed, (b) iq*, (c) id

*, (d) stator

phase current, ia (actual and command)………………………........... 40

xiii

Fig. 3.8: Comparison of speed error (= r* - r ): (a) conventional (b) NRM based

computation of id. ....................................................................................... 41

Fig. 3.9: Simulated responses of the IPMSM drive at very high speed (320 rad/s)

condition at rated load using the proposed NRM based id computation. ………...... 43

Fig. 3.10: Simulated responses of the proposed IPMSM drive for a step increase in

power in field weakening region……………………………………………….... 44

Fig. 3.11 Experimental responses of the IPMSM drive for a step increase in speed using

the proposed NRM based computation of id: (a) speed, (b) id. …………....... 45

Fig. 4.1 Simulink block diagram to get 3-phase currents from id and iq............... 49

Fig. 4.2 MATLAB/Simulink based built-in IPMSM model.............................. 49

Fig. 4.3 RKM (ode4) based calculation of IPMSM model: (a) ‘a’ phase stator current

(A), (b) steady-state 3-phase currents (A)................................................................ 50

Fig. 4.4 Result based on MATLAB/Simulink based built-in IPMSM model: (a) ‘a’

phase stator current (A), (b) steady-state 3-phase currents (A).............................. 50

xiv

List of Symbols

va, vb and vc a, b and c, phase voltages, respectively

ia, ib and ic Actual a, b and c, phase currents, respectively

ia*, ib

* and ic* Command a, b and c, phase currents, respectively

vd d-axis voltage

vq q-axis voltage

id d-axis current

iq q-axis current

Rs stator resistance per phase

Ld d-axis inductance

Lq q-axis inductance

s stator angular frequency (=Pr) or electrical frequency

r actual rotor speed (=dr/dt)

r* motor command speed

r error between actual and command speeds

r rotor position

P number of pole pairs

Te electromagnetic developed torque

TL load torque

Jm rotor inertia constant

Bm friction damping coefficient

m magnet flux linkage

Vm maximum stator phase voltage

Vm maximum stator phase voltage neglecting Rs

xv

Im maximum stator phase current

VB dc bus voltage for the inverter

Ts sampling period

xvi

List of Acronyms

AC Alternating current

DC Direct current

FW Flux-weakening

HPD High performance drive

IGBT Insulated gate bipolar transistor

IM Induction motor

IPMSM Interior permanent magnet synchronous motor

MTPA Maximum torque per ampere

NRM Newton-Raphson method

PI Proportional-integral

PID Proportional-integral-derivative

PM Permanent magnet

PMSM Permanent magnet synchronous motor

PWM Pulse width modulation

RKM Runge-Kutta method

VSI Voltage source inverter

xvii

Chapter 1

Introduction

1.1 Background of Motor

Advancement of modern societies is highly dependent on the development of

faster, efficient, and environmentally friendly motor drives because the motors consume

more than 50% of all the electrical power generated on the earth [1]. The motors are

used in all motions and locomotion of all devices such as electric vehicles, robotics,

rolling mills, machine tools, etc. This may be the most important factor behind today’s

high demand for more efficient motor control systems. Thus, the sophisticated and

precise computation of mathematical model of the motor is very important for high

performance motor drives. The development of electric motor has given us the most

efficient and effective means to do work known to the history. The electric motor

converts electrical energy into mechanical energy and the generator converts

mechanical energy in to electrical energy. Generally, there are two windings in a motor,

one receives electric energy and other gets electric energy by induction. Both of them

produce magnetic fluxes and due to the interaction of two fluxes rotating motion is

1

produced in the rotor. Over the years, electric motors have changed substantially in

design; however the basic principles have remained the same. In general electric motors

can be classified in to two major groups such as direct current (DC) and alternating

current (AC) motors.

1.1.1 Direct Current (DC) Motor

The modern DC motor was invented in 1873 by Belgian-born electrical engineer

Mr. Gramme. A simple permanent magnet DC motor is shown in Fig. 1.1 in which the

electric power in supplied from a battery to one winding and the main flux is supplied

by permanent magnets. Torque is produced according to Ampere’s Law, “If a current-

carrying conductor placed in a magnetic field, it experiences a force”.The name DC

motor comes from the DC electric power used to supply the motor. The DC motor was

in wide spread use in street railways, mining and industrial applications by the year

1900. However, the disadvantage of DC motors such as excessive wear in the electro-

mechanical commutator, low efficiency, fire hazards due to sparking, limited speed and

the extra room requirement to house the commutator, high cost of maintenance, became

evident and leads to further investigation in order to overcome these DC motor

disadvantages [2].

2

Fig. 1.1 Structure of a simple 2-pole PM DC motor.

1.1.2 Alternating Current (AC) Motor

Most AC motors being used today are so-called “induction motors” which are

the workhorse of the industry. In an induction motor (IM) a 3-phase AC supply is

applied to the stator and voltage is induced in the rotor according to Faraday’s law of

electromagnetic induction. The most common type of IM has a squirrel cage rotor in

which aluminum conductors or bars are shorted together at both ends of the rotor by

cast aluminum end rings as shown in Fig. 1.2. Due to the interaction of stator and rotor

fluxes rotating motion is produced. The rotor is connected to the motor shaft, so the

shaft will also rotate and drive the load. When a 3-phase supply is applied to the stator

winding a rotating magnetic flux/field is produced. The mechanical speed of the rotor is

always lower than the speed of the rotating magnetic flux (synchronous speed) by the so

called slip speed. The main advantages of IM are its lower cost, maintenance free

operation and greater reliability especially, in harsh industrial environment [2]. On the

contrary, IMs require very complex control scheme because of their nonlinear

relationship between the torque generating and magnetizing currents. In order to

3

overcome this problem, researcher developed another type of motor which is known as

synchronous motor.

Fig.1.2 Squirrel-cage induction motor.

1.1.3 Synchronous Motor

Synchronous motors utilize the same type of stator winding structure as IM and

which is either a wound DC field or permanent magnet rotor [2]. Synchronous motors

run at the synchronous speed which is the same as the supply frequency. In a

synchronous motor 3-phase AC supply is applied to the stator winding and a DC supply

is applied to the rotor winding which also have shorted squirrel cage bar. Initially, the

DC supply is not applied to the rotor and the motor starts like an IM when a 3-phase

supply is applied to the stator. When the speed of the motor is near synchronous speed

then the DC supply is applied to the rotor and the rotor magnetic field catches the stator

4

magnetic field and gets magnetically locked. Thus, the rotor rotates at synchronous

speed.

1.2 Background of the Permanent Magnet Synchronous

Motor (PMSM)

A permanent magnet synchronous motor (PMSM) is exactly similar to a

conventional synchronous motor with the exception that the field winding and DC

power supply are replaced by the permanent magnets (PMs). The PMSM has

advantages of high torque to current ratio, large power to weight ratio, high efficiency,

high power factor, low noise and robustness etc. Current improvement of PM motors is

directly related to the recent achievement in high energy permanent magnet materials.

Ferrite and rare earth/cobalt alloys such as neodymium-boron-iron (Nd-B-Fe),

aluminum-nickel-cobalt (Al-Ni-Co), samarium-cobalt (Sm-Co) are widely used as

magnetic materials. Rare earth/cobalt alloys have a high residual induction and coercive

force than the ferrite materials. But the cost of the materials is also high. So rare earth

magnets are usually used for high performance motor drives as high torque to inertia

ratio is attractive. Depending on the position of PM, there are different types of PMSM

which are discussed in the following subsection.

1.2.1 Classifications of PMSM

The performance of a PMSM drive varies with the magnet material, placement

of the magnet in the rotor, configuration of the rotor, the number of poles, and the

5

presence of dampers on the rotor [3]. Depending on magnet configuration, the PMSM

can be classified into following three types:

(a) Surface mounted PM motor: In this type of PM motor, the PMs are

typically glued or banded with a non-conducting material to the surface

of the rotor core as shown in Fig. 1.2(a). This type of motor is not

suitable for high speed.

(b) Inset type PM motor: In this type of PM motor, the PMs are typically

glued directly or banded with a non-conducting material inside the rotor

core as shown in Fig. 1.2(b). This type of motor is not suitable for high

speed.

(c) Interior type PM (IPM) synchronous motor (IPMSM): In an IPMSM, the

PMs are buried inside the rotor core as shown in Fig. 1.2(c). This is the

most recently developed method of mounting the magnets. Interior

magnet designs offer q-axis inductance (Lq) larger than the d-axis

inductance (Ld). The saliency makes possible a degree of flux

weakening, enabling operation above nominal speed at constant voltage

and should also help to reduce the harmonic losses in the motor. The

IPMSM has the advantages of smooth rotor surface, mechanical

robustness and a smaller air gap as compared to other types of PM

motors. Thus, this motor is more suitable for high speed operation and

hence, considered in this thesis. The parameters of the particular

laboratory 1 hp motor used in this thesis are given in Appendix A.

6

d-axisq-axis

(a)

(b)

(c)

Fig. 1.2: Cross section of a 4-pole: (a) surface mounted, (b) inset and (c) IPM type motor.

7

1.3 Motivation and Objectives

The permanent magnet synchronous motors (PMSM) get rapid industrial

acceptance because of their advantageous features such as high torque to current ratio,

high power to weight ratio, high efficiency, high power factor, low noise, and

robustness [6]. The advances in PM material quality and power electronics have boosted

the use of PMSM in electric motor drives for high performance applications such as

automotive, aerospace, rolling mills, etc. Among different types of PM synchronous

motors, the IPMSM shows excellent properties such as mechanically robust rotor, small

effective air gap, and rotor physical non-saliency [7]. The vector control technique is

widely used for high performance control of an IPMSM drive as the motor torque and

flux can be controlled separately in vector control [8]. Thus, the motor acts like a DC

motor while maintaining the general advantages of AC motor over DC motors.

Precise control of high performance IPMSM over wide speed range is an

engineering challenge. Fast and accurate speed response, quick recovery of speed from

any disturbances and insensitivity to parameter variations are some of the important

characteristics of high performance drive (HPD) system used in electric vehicles,

robotics, rolling mills, traction and spindle drives [6]. Some of the drive systems like

spindle drive and traction also need constant power operation [9]. The IPMSM drive

can be operated in constant power mode (above the rated speed) using the field-

weakening technique. In an IPMSM the direct control of field flux is not possible.

However, the field can be weakened by the demagnetizing effect of d-axis armature

reaction current, id [9]. Recently, the researchers [9-31] continue their efforts to develop

the IPMSM drive system incorporating the flux-weakening mode. The researchers

develop sophisticated control algorithms but often ignored the high precision

computation of the algorithm [9-31].

8

Electrical engineers often develop the mathematical model for voltage, current

and flux calculation of IPMSM drive but they simplify the equation for easier

computation. However, with the simplified equation the performance of the drive

deteriorates, which may not be suitable for HPD applications. Thus, the main objective

of the study is to investigate the numerical iterative computation methods (i.e., Newton-

Raphson and fourth order Runge-Kutta Methods) for high precision computation of

stator voltage/current and flux from mathematical model of IPMSM to improve the

performance of the drive. The particular objectives of this thesis are as follows:

to develop more accurate analytical model for flux control of IPMSM drive

based on d-axis stator current, id.

to incorporate the Newton-Raphson method (NRM) based iterative computation

method to calculate id for the closed loop vector control of IPMSM drive in

order to achieve high performance of the drive over wide speed range.

to investigate the performance of the IPMSM drive incorporating the proposed

NRM based calculation of id and it’s comparison with the conventional

simplified calculation of id.

to incorporate Runge-Kutta method (RKM) based iterative computation method

to solve the differential equation of IPMSM mathematical model and compare

the stator currents obtained from RKM with the MATLAB/Simulink library

model.

1.4 Research Methodology

The closed loop vector control of IPMSM drive is familiarized. For vector control

scheme the mathematical model of speed and current control algorithms are

familiarized. Then, the conventional flux control algorithm (i.e., computation of d-axis

9

stator current, id) is revisited. The performance (e.g. speed, current, etc.) of the IPMSM

drive with conventional flux control algorithm is investigated. After that a more

accurate analytical model for flux control of IPMSM is developed. The NRM is used to

compute the flux component of stator current, id. The performance of the IPMSM drive

is investigated with the proposed NRM based computation of id. These speed response

of the IPMSM drive with the proposed NRM based computation of id are compared with

the conventional computation of id in term of settling time, speed overshoot/undershoot,

steady-state error, etc.

Again, the fourth- order RKM (ode4) is applied to solve the set of first order

differential equations representing the mathematical model of IPMSM. The calculation

of motor stator currents using the ode4 are compared with those obtained from

MATLAB/Simulink library motor model.

1.5 Background of Numerical Computation Methodology

Various numerical methods e.g. Netwon-Rahpson and Runge-Kutta methods are

used for computation to find the solutions of different types of equations [4,5].

Electrical engineers develop mathematical model of electric motors for controlling

and/or efficiency optimization [5]. However, sophisticated computation of motor

mathematical model is often ignored by electrical engineers. Traditionally, they depend

on MATLAB/Simulink library and simplified equation for computation. Some

numerical methods used in this thesis are discussed below.

10

1.5.1 Newton-Raphson Method

In order to get more accurate value of id for proper flux control Newton-Raphson

Method (NRM) based numerical computation of id is proposed in this work.

The NRM is one of the most powerful, popular numerical methods and easy for

computer program to solve some root finding equations. As compared to other

numerical methods (e.g. Secant and Bisection) the NRM has some advantages such as,

it needs only one initial value, it's convergence is fast and it is easy to implement.

Therefore, in this thesis, the NRM will be used to solve some motor equations,

particularly, for high precision computation of flux for IPMSM. The NRM method is

illustrated below.

Let the function, f(x)=0, then according to NRM the roots of this equation can

be found using the iterative formulae,

xn+1=xn−f ( xn )

f '( xn ) , (1.1)

where, f΄(x) is the derivative of f(x).

Thus, the root of f(x) can be solved using the following steps.

1. Select: initial approximation, x1 = p0, tolerance (TOL), and maximum number of

iteration N0.

2. Do the following iterations using ‘for’ loop

For i=1:N0

calculate, x2=x1−

f ( x1)

f '( x1 )

3. If |x2-x1|<TOL, then the root of f(x) is: x2,

4. stop; otherwise,

5. set x1=x2; and repeat step 2 until step 4.

11

6. End of calculation.

1.5.2 Runge-Kutta Methods

In numerical analysis, the Runge–Kutta Methods (RKMs) are an important family of

implicit and explicit iterative methods for the approximation of solutions of ordinary

differential equations. These techniques were developed around 1900 by the German

mathematicians C. Runge and M.W. Kutta [5]. As compared to other methods, the

fourth order RKM (which is known as ode4) is well-known and very stable method to

solve for ordinary differential equations (ode) and this method is easy to implement in

computer program. Therefore, in this thesis, the fourth-order Runge-Kutta (RK4)

method will be used to solve some first ode representing the mathematical model of

IPMSM for high precision computation of its stator current. The RKM method is

illustrated below.

A first -order ordinary differential equation has the first derivative as its highest

derivative and it usually can be cast in the following form:

y'(x) = f (x,y) with initial condition y=y0 at x=x0 (1.2)

where, x is the independent variable, y(x) is the dependent variable and y'(x) is the first

derivative of y(x). The dependent variable at x+h, i.e. y(x+h), of equation (1.2), is given

by:

y(x+h) = y(x) + (k1 +2k2 + 2k3 + k4)/6 (1.3)

12

where,

k1 = hf(x0,y0)),

k2 = hf(x0+h/2,y0+k1/2),

k3 = hf(x0+h/2,y0+k2/2),

k4 = hf(x0+h,y0+k3),

and h is the fixed step-size.

The algorithm to solve Eqn. (1.3) using RK4 is:

To approximate the solution of the initial value problem y'(x) = f(x,y), a<x <b, y(0)=y0

at N+1 equally spaced numbers in the interval [a,b]:

INPUT end points a,b; integer N; initial condition y0:

OUTPUT approximation of y at the N+1 values of x.

Step 1: Set h=(b-a)/N;

x=a;

y=y0;

OUTPUT (x,y)

Step 2: For i=1,2,..,.N do Steps 3-5

Step 3: Set k1=hf(x,y);

k2=hf(x+h/2, y+k1/2);

k3=hf(x+h/2, y+k2/2);

k4=hf(x+h, y+k3);

Step 4: y=y+(k1+2k2+2k3+k4)/6; (Compute yi.)

x=a+ih. (Compute xi.)

Step 5: OUTPUT (x,y)

13

Step 6 STOP.

1.6 Organizations of the Thesis

The thesis consists of five Chapters. Chapter 1 presents the background about different

types of motors, particularly, IPMSM. Introduction about various numerical analysis

methods (e.g., NRM and RKMs) associated with the MATLAB programming. Then,

the motivation and the specific objectives of the current work are presented.

Chapter 2 provides a literature review on the IPMSM, its control techniques and

numerical analysis methods. Particularly, the mathematical model of IPMSM, vector

control strategy, speed and current control techniques are presented. Literature search

on flux control techniques for IPMSM and application of numerical analysis for motor

drives are also presented in this chapter.

In Chapter 3, the conventional and the proposed flux control techniques are

presented. For the proposed flux control, a more accurate analytical model of d-axis

stator current id incorporating stator resistance is developed. The NRM based

computation of id is also given in this chapter. The effectiveness of the proposed flux

control technique for IPMSM drive is investigated in both simulations and real-time.

Chapter 4 presents the application of 4 th order RKM to solve for d-q axis

components of stator current from the differential equations representing the

mathematical model of IPMSM. Then the a-b-c phase currents are calculated from d-q

axis currents. These currents are compared with the stator phase currents obtained from

MATLAB/Simulink built-in motor model.

Chapter 5 concludes the thesis indicating the achievements and limitations.

Finally, the future scope of the current work is also summarized.

14

Chapter 2

Literature Review

2.1 Mathematical Model of Interior Permanent Magnet

Synchronous Motor (IPMSM)

The IPMSM is similar to the conventional wire-wound excited synchronous

motor with the exception that the excitation is provided by the permanent magnets

instead of a wire-wound DC rotor field. The mathematical model of an IPMSM in the

d-, q-axis is given as [12],

vd=Ld

did

dt+Rs id−Pωr Lq iq (2.1)

vq=Lq

diq

dt+Rs iq+Pωr Ld id+Pωr ψm (2.2)

15

The electromagnetic developed torque in terms of electrical parameters is given by,

T e=3 P2

(ψm iq+(Ld−Lq ) id iq ) (2.3)

While, the electromagnetic developed torque in terms of mechanical parameters is given

by,

T e=T L+J m pωr+Bm ωr (2.4)

where, vd and vq are the d,q-axis voltages, Ld and Lq are the d,q-axis inductances, id and

iq are the d,q-axis stator currents, respectively; Rs is the stator resistance per phase, m is

the constant flux linkage due to rotor permanent magnet, r is the angular rotor speed, P

is the number of pole pairs of the motor, p is the differential operator (d/dt), and Te is the

developed electromagnetic torque, TL is the load torque, Bm is the viscous coefficient

and Jm is the inertia constant. The first term of (2.3) represents the magnet torque due to

the rotor permanent magnet flux m and the second term represents the reluctance torque

due to the complex interaction of d,q-axis currents and inductances of the IPMSM. The

complexity of the control arises due to the nonlinear nature of the torque equation (2.3)

because, Ld, Lq, id and iq are not constants. All these quantities vary during dynamic

operating conditions [9]. To make the torque equation linear and the control task easier,

usually id is set to zero [32]. However, in an actual IPMSM drive, it is inappropriate, the

assumption of id = 0 leads to erroneous results. If id = 0, the reluctance torque will be

zero and hence, the motor operates at below optimum torque condition. In this thesis id

is not considered zero. The value of id is calculated from iq maintaining the armature

voltage and current within the capacity of the motor and the inverter. This improves the

performance of the drive system as compared to the id = 0 technique as it can utilize the

16

PrLqiq+-Rs Ldid

vd

PrLdid+ -Rs Lq

vq

-+Prm

iq

reluctance torque below rated speed and operate the motor above rated speed by flux

weakening. The motor parameters used in the simulation are given in Appendix-A.

According to (2.1) and (2.2) the d,q-axis equivalent circuit models are shown in Fig. 2.1

(a) and (b), respectively.

(a)

(b)

Fig. 2.1. Equivalent circuit model of the IPMSM: (a) d-axis, (b) q-axis.

2.2 Vector Control Strategy for IPMSM Drive

The vector control means the control of both magnitude and phase angle of either

the motor voltage or current or both instead of their magnitudes only. As mentioned

earlier, the vector control technique is one of the most effective techniques for in high

performance AC motors drives. In the case of the permanent magnet synchronous motor

(PMSM), the torque eqn. (2.3) has two terms: the first term represents the magnet

17

torque produced by the permanent magnet flux m and the torque producing current

component iq; the second term represents the reluctance torque produced by the

complex interaction of inductances Ld and Lq and also the currents id and iq. The

excitation voltage due to permanent magnets, and the values of the inductances Ld and

Lq undergo significant variations in an interior type permanent magnet motor under

different steady state and dynamic loading conditions [9,33]. Thus, the complexity of

the control of the IPMSM drive arises due to the nonlinear nature of the torque eqn.

(2.3). In order to operate the motor in a vector control scheme avoiding the complexity,

id is set to zero. Then the torque equation becomes linear and is given by,

T e =

3 P2

ψm iq = K t iq (2.5)

where, the constant K t =

3 P2

ψm. Using phasor notations and taking the d-axis as a

reference phasor, the steady-state phase voltage Va can be derived from the steady-state

d -q axis voltage eqn. (2.1) and (2.2) as,

V a= vd + j vq

= R s Ia − ωs Lq iq + j ωs Ld id + j ωs ψm (2.6)

where, the phase current, I a= id + j iq (2.7)

In the case of the IPM motor, the d-axis current is negative and it demagnetizes the

main flux provided by the permanent magnets.

18

(a)

d-axis

q-axis

mid

iqIa

Lqiq

Ldido

-s Lqiq

jsm-jsLdidIaRs

Va vq

vd

d-axis

q-axis

m

Ia

Lqiqo

-sLqiq jsmIaRs

Va vq

vd

According to eqns. (2.6) and (2.7) the basic vector diagram of IPMSM is shown in Fig.

2.2 (a). The vector control scheme can be clearly understood by this vector diagram. It

is shown in the vector diagram that the stator current, Ia, can be controlled by

controlling the d- and q- axis current components. In the vector control scheme, when id

is set to zero then all the flux linkages are oriented in the d-axis as shown in Fig. 2.2 (b).

After setting id = 0, eqn. (2.3) shows that the torque is a function of only the quadrature

axis current component, iq and hence a constant torque can be obtained by ensuring iq

constant.

(b)

Fig. 2.2 Basic vector diagram of IPMSM: (a) general; (b) modified with id = 0.

19

+ r*

Current Controller

Vector rotator

SpeedController

VB

d/dt

ia* ib* ic*

r

iq* id*

ia ib

r-

ic

IPMSM Position sensorPWM Inverter

Base drive circuit

r

f(iq*, r)

The closed-loop vector control of voltage source inverter (VSI) fed IPMSM drive is

shown in Fig. 2.3. In this figure VB is the DC bus voltage for the inverter, which

supplies AC voltage to the motor with variable frequency and magnitude. The IPMSM

drive consists of the current controller and the speed controller. The speed controller

generates the torque command and hence the q-axis current command iq* from the error

between the command speed and the actual speed. As mentioned earlier, in the vector

control scheme traditionally, the d-axis command current id* is set to zero to simplify the

nonlinear dynamic model. If the flux control is needed the id should be calculated based

on some flux control algorithm

Fig. 2.3. Block diagram of the closed loop vector control of IPMSM drive.

20

of the IPMSM. The command phase currents ia*, ib

* and ic* are generated from the d,q

axis command currents using inverse Park’s transformation given in Eqn. (2.8) [34].

The current controller forces the load current to follow the command current as closely

as possible and hence forces the motor to follow the command speed due to the

feedback control. Therefore, in order to operate the motor in a vector control scheme the

feedback quantities will be the rotor angular position and the actual motor currents. In

the control scheme, the torque is maintained constant up to the rated speed, which is

also called the constant flux or the constant voltage to frequency ratio (V/f) control

technique.

[ia¿ ¿ ] [ib¿ ¿ ] ¿¿

¿¿ (2.8)

The (V/f) is maintained constant by using PWM operation of the VSI. The designs of

the speed controller, current controller and voltage source inverter to perform their

specific functions are given in the following sub-sections.

2.2.1 Speed Controller

The speed controller processes the speed error (r) between command and actual

speeds and generates the command q-axis current (iq*). The small change in speed r

produces a corresponding change in torque Te and taking the load torque TL as a

constant, the motor dynamic Eqn.(2.4) becomes,

21

ΔT e = J m

d ( Δωr)dt

+ Bm Δωr (2.9)

Integrating Eqn.(2.9) gives us the total change of torque as,

T e =K t iq = J m Δωr + Bm∫ 0

tΔωr (τ ) dτ

(2.10)

According to Eqn. (2.10), a proportional-integral (PI) algorithm can be used for the

speed controller which may be written as,

iq¿ = K p Δωr + K i∫0

tΔωr ( τ ) dτ

(2.11)

r = r*-r (2.12)

where, Kp is the proportional constant, Ki is the integral constant and r is the speed

error between the command speed, r* and the actual motor speed, r.

For discrete-time representation, Eqn. (2.11) can be differentiated and written in

discrete-time domain, respectively as,

diq¿

dt= K p

dΔωr

dt+ K i Δωr

(2.13)

iq¿ (k ) = iq¿ (k−1 ) + K p [Δωr ( k ) − Δωr (k − 1 ) ] +K iT s Δωr (k ) (2.14)

where, iq* (k) is the present sample of command torque, iq

*(k-1) is the past sample of

command torque, r(k) is the present sample of speed error and r(k-1) is the past

sample of speed error and Ts is the sampling period.

22

23

2.2.2 Current Controller and Voltage Source Inverter

The current controller is used to force the motor currents to follow the command

currents and hence forces the motor to follow the command speed due to the feedback

control. The outputs of the current controller are the pulse width modulated (PWM)

signals for the insulated gate bipolar resistor (IGBT) inverter switches. The voltage

source inverter (VSI) converts fixed DC voltage to a variable AC voltage for the motor

so that it can follow command speed with the required load. The current control

principle and PWM generation for the VSI can be found in [35].

2.3 Flux Control Methods for IPMSM Drives

In an IPMSM the flux cannot be controlled directly as the flux is supplied the

permanent magnets which are buried inside the rotor. The flux can only be controlled by

the armature reaction of the d-axis component of stator current, id. Traditionally, for

vector control of IPMSM drive id is set to zero so that the torque equation becomes

linear with iq and hence, the control task becomes easier. With id=0, motor cannot run

properly above rated speed as the flux remains fixed. If the flux is fixed, the voltage and

current exceed the rated limits of the motor with the speed since the voltage is

proportional to speed and flux. So, the flux control using id is needed to operate the

IPMSM above rated speed. Above the rated speed, the air gap flux is reduced by

demagnetizing effect of id so that the voltage and current don’t exceed the rated limits

although the speed exceeds the rated value. Moreover, with id 0, more torque can be

produced by the motor as the reluctance torque is utilized, which can be seen from Eqn.

(2.3).

24

Researchers reported some field weakening operations of IPMSM but the

computation of id is based on simplified equation which results in inappropriate control

of the flux and hence, non-optimum performance of the drive, particularly, above the

rated speed condition [14-31]. In this thesis, the Newton-Raphson method is used for

high precision computation of id without simplifying the equation. The flux control

techniques of IPMSM are discussed in detail in the next chapter.

2.4 Application of Numerical Analysis for Motor Drives

Traditionally the researchers in motor drives area depend on Matlab/Simulink

library as a dependable platform for computation. In order to reduce the computational

burden especially in real-time, sometimes they also simplify the equations even in

simulation so that they can resemble the simulation performance with the experimental

results. Recently, researchers looked into the application of numerical analysis methods

for modelling in motor drives [36-38]. In [36] authors used numerical analysis

technique for ultrasonic motor. Phyu et. al. used numerical modeling and analysis of

spindle motor for disk drive applications [37]. Han et. al. used numerical method for

brushless dc motor modeling [38]. None of the works are reported on the modeling of

IPMSM or its control. Moreover, the application of numerical analysis methods for

modeling of motor and/or control techniques are at its initial stage. There is a big scope

to investigate the application of numerical analysis methods for these applications

instead of readily available Matlab/Simulink or any other software.

25

2.5 Conclusion

In this Chapter first, mathematical model of IPMSM and vector control strategy

have been provided. Then, the speed and current control techniques have been briefly

described. A literature search on flux control techniques for IPMSM has been provided.

Finally, a literature search on application of numerical analysis for motor drives has

been provided.

26

Chapter 3

Flux Control Techniques for IPMSM

Drive

3.1 Introduction

The high performance motor drives used in robotics, rolling mills, machine tools,

electric vehicles, spindle drives, traction drives, etc. require low to high speed operation,

fast and accurate speed response, quick recovery of speed from any disturbances and

insensitivity of speed response to parameter variations. Some of the drive systems like

machine tool spindles and traction drives also need above rated speed (constant power)

operation. Despite many advantageous features, the precise torque and speed control of

IPMSM at high speed, particularly, above the rated speed, has always been a challenge

for researchers due to the nonlinearity present in the electromagnetic developed torque

because of magnetic saturation of the rotor core [33]. At high speeds, the voltage,

current and power capabilities of the motor exceed the rated limits. Consequently, the

27

magnetic saturation of the rotor core and shifting the operating point of permanent

magnet loading, and hence the variation of d, q axis flux linkages will be significant [9].

Thus, it severely affects the performance of the IPMSM drive at high speeds. For

IPMSM the direct flux weakening is not possible as the main flux is supplied by the

permanent magnets embedded in the rotor core. The IPMSM can be operated above the

rated speed by utilizing the demagnetizing effect of d-axis armature reaction current, id,

which makes the flux weakening operation [14-31]. Moreover, below the rated speed

with proper control of id the reluctance torque can also be utilized [6]. So, without

controlling the flux, the additional advantages of IPMSM over other PM motors cannot

be achieved. Usually, for vector control of IPMSM id is set to zero in order to linearize

the IPMSM model and to make the control task simpler [35]. Researchers reported

some field weakening operations of IPMSM but the computation of id is based on

simplified equation which results in inappropriate control of the flux and hence, non-

optimum performance of the drive, particularly, above the rated speed condition [14-

31]. In this work, the Newton-Raphson method is used for high precision computation

of id without simplifying the equation [39]. In order to control the net air-gap flux of

IPMSM, the d-axis stator current id is controlled according to the maximum torque per

ampere (MTPA) operation in constant torque region (below the rated speed) and the

flux-weakening (FW) operation in constant power region (above the rated speed).

3.2 Conventional Flux Control of IPMSM

As per phasor diagram Fig. 2.2(a)) the net air-gap flux linkage, o can be defined as,

ψo = √ψd2+ψq

2 (3.1)

where, d and q are d,q axis flux linkages, respectively, which can be defined as,

28

ψd = Ld id+ψm (3.2)

ψq = Lq iq (3.3)

Since id is negative for an IPMSM, it can be seen from Eqns. (3.1) and (3.2) that air-gap

flux linkage can be reduced by varying id. This is called demagnetizing effect of id.

Traditionally, id is set to zero to make the control task easier [29]. With id=0, motor

cannot run properly above rated speed as the flux can’t be controlled. So, the flux

control using id is needed to operate the IPMSM above rated speed within the rated

voltage and current capacities of the motor. Moreover, with id 0, more torque can be

produced by the motor as the reluctance torque is utilized.

3.2.1 Conventional Flux Control below Rated Speed – Maximum

Torque per Ampere (MTPA) Control

Referring to the phasor diagram of Fig. 2.2(a), the stator phase voltage (V a ) and

current ( I a ) can be related to the d-q axis voltages and currents as,

V a = vd + jvq (3.4)

I a = id + jiq (3.5)

The maximum value of stator phase voltage and current are considered as, Vm and Im,

respectively. Below the rated speed, with the assumption of keeping the absolute value

of stator current ( I a ) constant, id can be calculated in terms of iq for MTPA control. This

is obtained by differentiating electromagnetic developed torque equation (2.3) and

setting it to zero to get the maximum torque as [12],

29

dT e

diq= d

diq( 3 P

2 [ψmiq + (Ld − Lq ) id iq] )=0

(3.6)

and, I a =√( id )2+( iq )

2 (3.7)

After solving id can be obtained as,

id =ψm

2(Lq−Ld )− √ ψm

2

4 (Lq−Ld )2+ iq

2

(3.8)

In real-time, implementatation of the drive system becomes complex, and it

overburdens the DSP with expressions in equation (3.8). In order to solve this problem,

usually a simpler relationship between d- and q-axis currents, which is obtained by

expanding the square root term of equation (3.8) using Maclaurin series expansion as

(3.9). The values of the motor parameters in (3.8) are given in Appendix-A. In this case

only first two terms of series expansion are considered for simplicity.

(3.9)

This is the equation which is used to get id below rated speed.

3.2.2 Conventional Flux Control above Rated Speed – Field

Weakening (FW) Control

As the voltage is proportional to speed and flux, above rated speed the voltage and

hence the currents exceed the rated limit if the flux remains constant. In order to operate

the motor within the rated capacities the flux must be weakened. As discussed earlier,

for an IPMSM the flux can be weakened by the demagnetizing effect of id.

Conventionally, above the rated speed id is obtained from Eqns. (2.1), (2.2), and (3.4)

30

neglecting the stator resistance (Rs) and assuming the stator voltage constant at its

maximum value (Vm) as follows [12].

Neglecting Rs, from Eqn. (2.1) we get,

vdo = −Pωr iq Lq (3.10)

Neglecting Rs, from Eqn. (2.2) one can get,

vqo = Pωr Ld id+Pωr ψm (3.11)

From Eqn. (3.4),

V m' =√(vdo )

2+( vqo)2

(3.12)

From Eqns. (3.10)-(3.12) id can be obtained as,

id=−ψm

Ld+

1Ld √ V

m2'

P2 ωr2 −Lq

2 iq2

(3.13)

This equation is used to calculate id above the rated speed. Further to reduce the

computational burden, Maclaurin series expansion (only first one term is considered)

can be applied to Eqn. (3.13), which yields [30]:

id ≈ −10 . 328+2356 .27ωr

(1−316 .57 x10−9 ωr2 iq

2 ) (3.14)

Above the rated speed, the voltage remains constant as the magnet flux is weakened by

the d-axis armature reaction current, id and hence, the power remains constant in this

region. The typical torque-speed characteristic curve of a motor drive is shown in

Fig.3.1. Equation (3.13) represents an ellipse in the d-q plane, which indicates that an

increase in rotor speed results in smaller ranges for the current vector as shown in

Fig.3.2. By appropriately controlling id, the amplitude of the terminal voltage is adjusted 31

Speed

Torque

r,rated r,max

MTPA Region FW Region

Constant torque region Constant power region

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-18

-16

-14

-12

-10

-8

-6

-4

-2

0

2

Ia 188213238

Current iq, A

Current id, A

to Vm. Above the base speed i.e., in constant power region, the voltage remains constant

as the magnet flux is weakened by the armature reaction of id in order to decrease the

total air gap flux. The flux-weakening control not only extends the operating limits of

IPMSM drive but also relieves the current controllers from saturation that occurs at high

speeds. Saturation of the current controllers occur at high speeds for a given torque,

when the motor terminal voltage approaches Vm, which may cause instability of the

drive for id=0 control.

Fig. 3.1 Typical torque-speed characteristic curve of a motor drive.

Fig. 3.2: Locus of stator current Ia at different speeds staring from 188 rad/s to 363 rad/s

in a step of 25 rad/s.

32

3.3 Proposed Flux Control Techniques

In section 3.2 the current id is calculated based on Maclaurin series in order to

reduce the computational burden so that it becomes suitable for real-time

implementation. However, some approximation is taken in Maclaurin series to compute

id, which results in an improper value of the flux. Moreover, in order to simplify the

calculation most of the reported works neglected the stator resistance to calculate id in

the field weakening region [5-27]. Thus, the motor performance gets affected especially

at high speed condition. Therefore, in this thesis first, a more accurate analytical model

of id is developed in the FW region including the stator resistance. Then, the NRM

based numerical computation of id is used so that the flux calculation becomes more

accurate and hence, the motor runs smoothly above rated speed conditions.

33

3.3.1 Proposed Flux Control below Rated Speed

Below the rated speed for NRM based computation of id, Eqn. (3.8) can be

rewritten as,

f ( id)= {id2−

ψm id

(Lq−Ld)− iq

2}=0 (3.15)

According to NRM,

id (n+1 )= id (n )−f ( id )

f ' ( id )|id=id(n)

(3.16)

The Matlab program to solve for id below rated speed using (3.16) is shown in

Appendix-B. Below the rated speed the calculation of id does not show that much

difference between NRM based computation using Eqn. (3.16) and the conventional

computation using Eqn. (3.14). That is why, further analysis for id calculation below the

rated speed is not provided. The flux calculation above the rated speed is more

important than below the rated speed and hence, the numerical computation of id above

the rated speed is provided below.

3.3.2 Proposed Flux Control above Rated Speed

Above the rated speed for NRM based computation of id without neglecting stator

resistance, in steady-state the FW method can be re-evaluated from Eqn. (2.1), (2.2) and

(3.4) as follows [40]:

vd = R sid−Pωr iq Lq (3.17)

vq = R s iq+Pωr id Ld+Pωr ψm (3.18)

V m =√(vd )2+( vq )

2 (3.19)

⇒(Pωr ( Ld id+ψm)+R s iq )2+(R s id−Pωr Lq iq)

2=V m2

(3.20)

34

So, f ( id )= (Pωr (Ld id+ψ f )+Rs iq)2+(R s id−Pωr Lq iq )

2−V m2 =0 (3.21)

where, Vm is the maximum stator phase voltage without neglecting stator resistance.

From Eqns. (3.16) and (3.21), for certain values of iq and r, id can iteratively calculated

as,

(3.22)

Based on the iteration the value of id is calculated if the consecutive values converges

with the precision of =10-5. That means if |id (n+1 )− id (n )|≤ε then id=id(n+1). The

Matlab program to solve for id above rated speed using (3.22) is shown in Appendix-C.

First the value of id is calculated offline at each speed using the NRM and then a

polynomial of id as a function of speed is developed, which is used for flux weakening

algorithm.

Above the rated speed the variations of the proposed id with different speeds are

shown in Fig. 3.3. The values of id calculated based on simplified Eqn. (3.14) are also

shown in this figure. It is found that there is a significant difference between the two

values that causes the enhancement in the performance. It is also found that id calculated

based on Eqn. (3.14) is positive for speeds above but near rated speed (188.5 rad/s =

1800 rev/min(rpm)), which is not making the field weakening operation. Whereas, the

NRM based numerical computation of id incorporating the stator resistance (Rs) using

Eqn. (3.22) is always negative and hence, ensuring the proper field weakening

operation. Using the cubic polynomial curve fitting method the equation of id (computed

based on NRM) as a function of speed is determined as,

id = 0.0004r3 - 0.0129r

2 + 15.829r (3.23)

35

180 200 220 240 260 280 300 320 340-4

-3

-2

-1

0

1

2

id based on simplified equation (3.14)

id based on NRM incorporating Rs

Speed, rad/s

id, A

180 200 220 240 260 280 300 320 340 360-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

O - id calculated based on NRM incorporating Rs

id based on polynomial (3.23) id

Speed, rad/svd

id, A

This equation is used to calculate id for field weakening operation of IPMSM

above rated speed. The curve fitting for higher degree polynomial exhibits bad

conditioned. Moreover, unnecessary it will increase the computational burden. The

curve fitting equation was checked with the earlier data and the plot is shown in Fig.

3.4. It is found that the actual id matches almost perfectly with the values obtained from

polynomial (3.23). Thus, Eqn. (3.23) is used to calculate id above rated speed.

Fig. 3.3: Variation of id with speed, r.

Fig. 3.4: Comparison of curve fitting polynomial with actual id.

3.4 Simulation Results and Discussions

A closed loop vector of IPMSM drive incorporating the proposed control

algorithms mentioned earlier is developed in Matlab/Simulink [41]. The Simulink block

36

diagram for the proposed IPMSM drive and the reference current generator subsystem

are shown in Fig. 3.5(a) and 3.5(b), respectively. The flux control algorithm blocks

(MTPA and Flux_weaken), which are the main concern of this thesis, are shown in Fig.

3.5(b). The inputs of the command current generator subsystem shown in Fig. 3.5(b) are

command speed and the feedback actual speed of the motor. The outputs are the

command currents IAC, IBC and ICC. The speed controller is shown as proportional-

integral-derivative (PID) but the derivative part is considered zero so that it acts like PI

controller. The other Simulink subsystems of Fig 3.5(a) are given in Appendix-D. The

performance of the proposed drive system is investigated in simulation at different

operating conditions. Sample simulation results are presented in Figs. (3.6)-(3.10).

The simulated transient responses of the IPMSM drive for a step change of

command speed are shown in Figs. 3.6 and 3.7 for the conventional and proposed

calculation of id [8], respectively. It is clearly seen from Fig. 3.7(a) that the proposed

NRM based numerical computation of id ensures proper flux weakening operation and

hence there is no overshoot in the speed response when the speed increase from

188.5rad/s (=1800 revolution/min (rpm)) to 250 rad/s (=2387 rpm). Whereas, it is seen

from Fig. 3.6(a) that the speed response suffers from big overshoot due to the simplified

calculation of id. Therefore, the incorporation of the stator resistance to calculate id

based on NRM is justified. The torque component of the current (iq) decreases above

rated speed since the torque decreases in the field weakening region (Figs. 3.6(b) and

3.7(b)). For field weakening operation id is becoming more negative when the speed

goes above rated speed, which is shown in Fig. 3.7(c). It is seen from Fig. 3.6(d) that the

actual motor current cannot follow the command current at 250rad/s for conventional id

control as the current controller gets saturated due to improper flux control. Whereas, it

is seen from 3.7(d) that for the proposed id control actual motor current can follow the

command current in steady-state without any error.

37

iq

To Workspace9

ib

To Workspace8

ic

To Workspace7

t

To Workspace6

wrc

To Workspace5

ICC

To Workspace4

IBC

To Workspace3

IAC

To Workspace2

NA

To Workspace11

id

To Workspace10

ia

To Workspace1

wr

To Workspace

Switch1

Scope

wrc

cos_the

wr

sin_the

IAC

IBC

ICC

Ref_curr

iq

id

wr

th_e

Motor_out

NA

NB

NC

v qs

v ds

Inverter

IAC

ia

IBC

ib

ICC

ic

NA

NB

NC

Hys_cnt

sin(u)

Fcn2

cos(u)

Fcn1

wr

v q

v d

iq

Curr_trans

cos_the

v qs

sin_the

v dsv d

Coor_trans

250

Constant6

188.5

Constant4

0

Clock1

0

Clock

cos_the

iq

id

sin_the

ia

ib

ic

Actual_curr

3ICC

2IBC

1IAC

costheta

iq

sintheta

id

iac

ibc

icc

dq-abc trans

Switch

Sum1

PID

PID Controller(with Approximate

Derivative)

iqr* idr*

MTPA

iqr*

wr

idr*

Flux_weaken

-K-

(1/K)T*

4sin_the

3wr

2cos_the1wrc

Fig. 3.5. (a) Simulink block diagram for overall control system of IPMSM drive, (b)

Reference current generator subsystem.

38

(a)

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

250

300

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Current id*, ACurrent iq*, A

Speed, rad/s

(a)

(b)

(c)

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75-8

-6

-4

-2

0

2

4

6 (d)

Current ia, A

Time, s.

Actual motor speed

Reference speed

Fig. 3.6: Simulated transient responses of the drive for step change of speed at rated

load using the conventional computation of id [12]; (a) speed, (b) iq*, (c) id

*, (d) stator

phase current, ia (actual and command).

39

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-8

-7

-6

-5

-4

-3

-2

-1

0

id*, A

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

(b)

Speed, rad/s

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

250

300

Actual speed

(a)

Reference speed

(c)iq*, A

0.4 0.45 0.5 0.55 0.6 0.65 0.7

-6

-4

-2

0

2

4

6

Time, s

ia, A

(d)

Fig. 3.7: Simulated transient responses of the IPMSM drive for step change of speed at

rated load using the proposed NRM based id computation; ; (a) speed, (b) iq*, (c) id

*, (d)

stator phase current, ia (actual and command).

40

Fig. 3.8 shows the comparative speed errors for conventional and proposed NRM based

computation of id. The performance comparison of the IPMSM drive with conventional

and proposed id computation is summarized in Table-I. It is found that the conventional

drive has 10.8 % overshoot and the proposed drive has no overshoot at high speed (250

rad/s) condition. Moreover, the proposed drive has less settling time than the

conventional one.

Fig. 3.8: Comparison of speed error (= r* - r ): (a) conventional (b) NRM based

computation of id.

Table-I: Performance comparison of conventional and proposed computation of id

Conventional id computation

Proposed NRM based id computation

188.5 rad/s 250 rad/s 188.5 rad/s 250 rad/s

% Speed overshoot ((/ r

*)*100%)4.51 10.8 4.51 0

Settling time (s) 0.2 s 0.2 s 0.2 s 0.1 s

41

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-50

0

50

100

150

200

Time, s

(b)

Spe

ed e

rror

(r

ad/s

) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-50

0

50

100

150

200

Time, s

Spe

ed e

rror

(r

ad/s

)

(a)

Fig. 3.9 shows the response of the IPMSM drive at very high speed (320 rad/s

3056rpm) using the proposed NRM based id computation. The motor can follow this

high command speed only due to the proper calculation of id. With conventional

simplified computation of id, it was not possible to run the motor at such high speed

condition. Thus, the proposed technique extends the operating speed region of the

motor. As the speed increases, the frequency of the stator current increases, which is

shown in Fig. 3.9(c). The 3-phase currents shown in Fig. 3.9(d) indicate the balanced

operation of the motor.

A load disturbance is also applied to the motor and the corresponding responses are

shown in Fig. 3.10. The load is increased at t=0.8 s while the motor is running at 250

rad/s. The d,q axis currents adjusted with the new load condition as the id decreases and

iq increases further at t=0.8s. It is found that the motor can handle the load disturbance

while running above rated speed condition. Therefore, the proposed NRM based

numerical computation of id provides proper field weakening operation and hence the

drive can handle the uncertainties such as sudden change in reference speed and load

while running at high speed (above rated speed) condition. It has been found from the

results that the IPMSM drive with proposed calculation of id provides better response as

compared to the conventional calculation of id. Thus, the proposed method could be a

potential candidate for real-time field weakening operation of IPM motor.

42

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

250

300

350

0.605 0.61 0.615 0.62 0.625-3

-2

-1

0

1

2

30 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-8

-6

-4

-2

0

2

4

6

8

10

Time, s

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-30

-25

-20

-15

-10

-5

0

5

10

15

iq*

id*

Actual speed

Reference speed

(a)

(b)

(c)

(d)

Speed, rad/sd/dt

iq*, id* , A ia, AEncoder

ia, ib, ic, A

Fig. 3.9: Simulated responses of the IPMSM drive at very high speed (320 rad/s)

condition at rated load using the proposed NRM based id computation: (a) speed, (b)

command d-q axis currents id* and iq

*, (c)’a’ phase stator current, (d) steady-state 3-

phase stator currents of the motor.

43

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

250

300

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-12

-10

-8

-6

-4

-2

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

(a)

(b)

(c)

Current iq*, ACurrent id*, A

Speed, rad/s

Time, s

Fig. 3.10: Simulated responses of the proposed IPMSM drive for a step increase in

power in field weakening region: (a) speed, (b) command d-axis current id*, and (c)

command q-axis current iq*.

44

X-scale: 500ms/div

Y-scale: 40 rad/sec/div

100

200 250

188.5Y-scale: 25 rad/s/divX-scale: 0.5 s/div

Actual speed

Command speed

(b)

id, A

3.5 Experimental Test of the Proposed Field Weakening

The proposed id control method was tested in real-time at “Lakehead University

Power Electronics and Drives Laboratory” [26]. A sample experimental result is given

below. The experimental speed and the corresponding d-axis current responses for the

proposed NRM based IPMSM drive at a step change in command speed from 188.5

rad/s to 250 rad/s are shown in Fig. 3.11. It is found that the motor follows the above

rated command speed (250 rad/s) smoothly without any steady-state error, overshoot

and undershoot. It is seen from Fig. 3.11(b) that the d-axis current becomes more

negative for proper field weakening operation when the speed increases above rated

speed. Thus, the model developed for id and it’s computation using NRM are also

verified in real-time.

Fig. 3.11 Experimental responses of the IPMSM drive for a step increase in speed using

the proposed NRM based computation of id: (a) speed, (b) id.

45

(a)

3.6 Conclusion

In this Chapter at the beginning the necessity of flux control technique has been

discussed. Then, the conventional flux control technique and the proposed flux control

algorithms have been discussed in detail. For the proposed flux control, a more accurate

analytical model of d-axis stator current id incorporating stator resistance has been

developed. The NRM based computation of id has also been presented. The simulation

results of the proposed flux control technique for IPMSM drive at different operating

conditions have been presented. The effectiveness of the proposed flux control

technique has also been verified in real-time.

46

Chapter 4

Application of Runge-Kutta Method for

Solution of IPMSM Mathematical Model

4.1 Introduction

Traditionally, the MATLAB/Simulink library built-in motor model is used for

drives simulation [12,30]. Researchers usually overlooked about the accuracy of the

simulation performance of the drive caused by MATLAB/Simulink built-in motor

model. Therefore, it is necessary to investigate the use of numerical methods for the

solution of motor mathematical model. In this chapter the fourth order Runge-Kutta

method (RK4) is applied to solve the 1st order differential equations representing the

mathematical model of IPMSM. Then the motor currents obtained from RK4 are

compared with those of obtained from MATLAB/Simulink built-in motor model. The

limitations of the built-in motor model is also discussed in this Chapter.

47

4.2 Modelling of IPMSM

In order to apply RK4 the current based mathematical models of IPMSM are used,

which are given by,

Ld

did

dt=vd−R s id+Pωr Lq iq

(4.1)

Lq

diq

dt=vq−R s iq−Pωr Ld id−Pωr ψm (4.2)

For the solutions of (4.1) and (4.2) the values of vq and vd are considered as +150V and -

150V, respectively, which are corresponding to the rated values of stator voltages. After

putting all the motor parameters and rated speed values Eqns. (4.1) and (4.2) becomes,

did

dt=−3534 .401−45 .48 id+7066.83 iq=f 1( t , i d , iq ) (4.3)

diq

dt=397 . 78−24 .26 iq−201. 03 id= f 2( t , i d , iq ) (4.4)

Utilizing 4th order RKM id and iq are solved from these equations. The corresponding

Matlab codes are shown in Appendix E.

4.3 Simulation Results of the RKM and MATLAB/Simulink

Based Motor Model

The 3-phase stator currents (ia*, ib

* and ic*) are calculated from the d-q axis currents,

id and iq, obtained from Rk4 using inverse Parks’ transformation as given in Eqn. (2.8).

The Simulink block diagram to implement Eqn. (2.8) is shown in Fig. 4.1. Then the

stator ‘a’ phase current obtained from RK4 is compared with the one obtained from the

Simulink built-in machine model, which is shown in Fig. 4.2 [31].

48

te

torque

wr

speed

VDC

g

A

B

C

+

-

Universal Bridge

z

1

2.25

Load (Nm)

Tm

mABC

I nterior Permanent MagnetSynchronous Machine

Iabc*

Iabc

Pulses

Current RegulatorCommand currents from speed controller

Fig. 4.1 Simulink block diagram to get 3-phase currents from id and iq.

Fig. 4.2 MATLAB/Simulink based built-in IPMSM model.

The comparative results are shown in Figs. 4.3 and 4.4 for RKM and

MATLAB/Simulink, respectively. It is found that the steady-state current responses are

almost similar. There is a little bit difference in transient state (beginning part) as Fig.

4.3 was obtained in open loop just for the sake of testing the RKM and Fig. 4.4 was

obtained under closed loop condition. If the RKM is applied in closed loop condition,

49

iq

3icc

2 ibc

1 iac

id

To Workspace4

iq

To Workspace2

Sum7

Sum6

Sum3

Sum2

Product6

Product5

Product1

Product

1/2

Gain3

1

Gain1

-K-

1.732

4id

3sintheta

2iq

1costheta

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-12

-10

-8

-6

-4

-2

0

2

4

6

Time, s

(a)

ia, A

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-8

-6

-4

-2

0

2

4

6

8

10

12

(a)

ia, A

Time, s

0.4 0.41 0.42 0.43 0.44 0.45 0.46-5

-4

-3

-2

-1

0

1

2

3

4

5

(b)ia, ib, ic, A

Time, s0.4 0.41 0.42 0.43 0.44 0.45 0.46

-5

-4

-3

-2

-1

0

1

2

3

4

5

Time, s

(b)

ia, ib, ic, A

the transient response will also be same. Therefore, the machine model available in

MATLAB/Simulink can be used without any hesitation.

50

Fig. 4.3 RKM (ode4) based calculation of IPMSM model: (a) ‘a’ phase stator current (A), (b) steady-state 3-phase currents (A).

Fig. 4.4 Result based on MATLAB/Simulink based built-in IPMSM model: (a) ‘a’ phase stator current (A), (b) steady-state 3-phase currents (A).

4.4 Conclusion

The RKM based solution of IPMSM mathematical model has been presented in this

Chapter. From comparative results it is found that the MATLAB/SIMULINK library

built-in motor model can be used for general purpose as it provides acceptable results.

The built-in motor model can save lot of time for the researcher to simulate the drive's

performance. However, if more flexibility is required to change/include the motor

parameters (e.g., the effect of magnetic saturation, core loss, etc.) values the

mathematical model of the motor should be developed in software as these options are

not available in Simulink library built-in machine model.

51

Chapter 5

Conclusions

5.1 Summary of the Thesis

Based on the review of various electric motors in Chapter 1, it is concluded that the

interior permanent magnet synchronous motor (IPMSM) can be used to meet the criteria

of variable speed high performance electric motor drives. Problems of controlling the

IPMSM above rated speed have been identified. It has been found that the high

precision computation of flux is needed to run the IPMSM properly above rated speed.

The numerical analysis techniques such as Newton-Raphson and Runge-Kutta methods

have been proposed for performance investigation of the IPMSM.

The existing literature about mathematical model of the IPMSM, vector control

technique, speed controller, current controller and flux control techniques have been

reviewed in Chapter 2.

In Chapter 3 the flux control techniques of the IPMSM to extend the operating

speed range has been discussed. The techniques include the maximum torque per

ampere operation in the constant torque region (i.e., below the base speed) and the flux-

weakening operation in the constant power region (i.e., above the base speed). The flux

52

of IPMSM can be controlled by demagnetizing effect of d-axis stator current, id. A new

analytical model of id incorporating the stator resistance has been developed in this

Chapter. Then the NRM has been applied for high precision computation of id. The

performance of the proposed NRM based id control technique has been investigated in

simulation. A comparison has also been made between the proposed and the

conventional id computation based IPMSM drive. The proposed id computation

technique has been found superior to the conventional id control technique over a wide

speed range. Finally, the performance of the proposed id computation based IPMSM

drive has been verified in real-time.

In Chapter 4 the fourth order Runge-Kutta method (RK4) has been applied to solve

the ordinary differential equation representing the mathematical model of IPMSM. The

stator currents of the IPMSM have been computed based on RK4 and compared with

those of MATLAB/Simulink built-in motor model. It has been found that the current are

almost similar. However, it has been proposed to solve the mathematical model using

RK4 if greater flexibility such as to incorporate magnetic saturation, parameter variation

and iron loss resistance is needed to investigate the performance of the drive in

simulation.

5.2 Major Achievement of the Thesis

The achievement of the current work can be summarized as follows:

(1) In this work, the Newton-Raphson based numerical method has been used for high

precision computation of flux component of stator current, id.

53

(2) A new analytical model of id incorporating the stator resistance has been developed.

The calculated id provides proper flux weakening operation of IPMSM drive to

operate the motor above the rated speed over a wide speed range.

(3) The performance of the drive system with the proposed NRM based computation of

id has been investigated in simulation using Matlab/Simulink at different operating

conditions. The effectiveness of the proposed NRM based id computation has also

been tested in real-time.

(4) The performance of the IPMSM drive with the proposed NRM based computation of

id has also been compared with the conventional [12] simplified computation of id.

The IPMSM drive with proposed calculation of id provides better speed response as

compared to the conventional calculation of id. Thus, the proposed method could be a

potential candidate for industrial field weakening operation of IPMSM.

5) The fourth order Runge-Kutta method has been used to solve the mathematical model

of IPMSM and then the performance has been compared with the Simulink built-in

motor model. Through this investigation researcher can find the direction if more

flexibility in motor model is required for simulating the motor performance at

different operating conditions.

5.3 Future Scope of the Current Work

Other numerical methods may be applied for flux control and the results can be

compared with the current work.

Mathematical model of intelligent (e.g., fuzzy logic, neural network, etc.) flux

control scheme can be developed based on motor operating conditions (e.g.,

voltage, current and load) so that the model will be independent of motor

parameters (e.g., inductance and resistance). Then, the performance can be

54

compared with the current work.

Fourth order Runge-Kutta Method may be applied to solve the mathematical

model of IPMSM incorporating magnetic saturation and iron loss. Higher order

Runge-Kutta Methods may be also applied to investigate the solution of IPMSM

mathematical model and then it can be compared with the MATLAB/Simulink

motor model.

55

56

List of Papers Published from this Work

1. S. Pervin, Z. Siri and M. Nasir Uddin, “Newton-Raphson Based Computation of id

in the Field Weakening Region of IPM Motor Incorporating the Stator Resistance to

Improve the Performance”,Proceedings of 47thIEEE (Institute of

Electrical and Electronic Engineers) Industry Applications Society

(IAS) Annual Meeting, Oct. 2012, Las Vegas, USA, pp.1-6.

2. S. Pervin, M. Nasir Uddin and Z. Siri, “Improved Dynamic

Performance of IPMSM over Wide Speed Range Based on

Numerical Computation of id in the Field Weakening Region”,

Journal of International Review of Modeling and Simulations

(IREMOS), Praise Worthy Prize publishing house, Italy, vol.5, no. 5,

Oct. 2012, pp. 2042-2048. (SCOPUS cited)

3. S. Pervin, “Numerical Methods Based Computation and Analysis of Interior

Permanent Magnet Synchronous Motor Drive”, Colloquiums series, Institute of

Mathematical Sciences, University of Malaya, 27 June 2012.

57

58

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63

APPENDIX-A: IPMSM PARAMETERS

Appendix-B: (Application of NRM to solve for id below rated speed)

% Name: Shahida Pervin% Matric #: SGP 110002% Supervisor: Dr. Zailan Siri% Newton-Rephson Method for PM motor flux calculation below rated speed% id calculation if w,wrated; f(id)= id^2-xf*id/(2*(Ld-Lq))-iq^2Ld=0.04244;Lq=0.07957;iq= 2.45;xf=0.314;id1=0;for i=1:10id2=id1- (id1^2-(xf*id1)/(2*(Lq-Ld))-iq^2)/(2*id1-xf/(2*(Lq-Ld)));if abs(id2-id1)<=0.00001 output=id2endid1=id2;end

64

Motor rated power 1 hp

Rated voltage 208 V

Rated current 3 A

Rated frequency and phase 60 Hz, 3-phase

Pole numbers 4 (pole-pair, P=2)

d-axis Inductance, Ld 42.44 mH

q-axis Inductance, Lq 79.57 mH

Damping coefficient, Bm .0008 Nm/rad/sec.

Motor inertia, Jm .003 kg-m

Stator resistance, Rs 1.93 Ohm

Magnet flux linkage, m 0.314 V/(rad/s)

Magnet type Samarium cobalt

Appendix-C: (Application of NRM to solve for id above rated speed)

% Name: Shahida Pervin% Matric #: SGP 110002% Supervisor: Dr. Zailan Siri% Newton-Rephson Method for PM motor flux calculation above rated speed% id calculation if w>wrated;clearP=2;R=1.93;Ld=0.04244;Lq=0.07957;xf=0.313;Vm=240;wr=300;iq=424/(wr*.939);%wr=188:12.5:363;%NN=length(wr);%for ii=1:NNid1=0; for i=1:10id11=(P*wr*(Ld*id1+xf)+R*iq)^2+(R*id1-P*wr*Lq*iq)^2-Vm^2;

id22=2*R*(R*id1-P*wr*Lq*iq)+2*P*wr*Ld*(P*wr*(Ld*id1+xf)+R*iq); id2=id1-id11/id22; if abs(id2-id1)<=0.00001 output=id2;endid1=id2;endid=output%id(ii)=output;%end%plot(id,wr)

65

Appendix-D: Simulink Subsystems

The Simulink block diagram for the proposed IPMSM drive and the reference

current generator subsystem were shown in Fig. 3.5(a) and 3.5(b), respectively. The

details of the other subsystem blocks for the Simulink schematic shown in Fig. 3.5 (a)

are presented in this appendix. The inputs of the command current generator subsystem

shown in Fig. 3.5(b) are command speed and the feedback actual speed of the motor.

The outputs are the command currents IAC, IBC and ICC. The command currents act as

inputs to the current controller subsystems (D.1). In current controller subsystems, the

relays are used to represent the hysteresis band. The actual motor currents (ia, ib, ic) also

act as inputs to this subsystem. The outputs of this subsystem are the logic signals NA,

NB and NC, which are used to turn the inverter switches on and off. The inputs of the

inverter subsystem (D.2) are logic signals NA, NB, NC and the dc bus voltage VB. The

outputs of this subsystem are the d-q axis voltages in the stationary reference frame. The

coordinate transformation subsystem (D.3) transforms d-q axis voltages from stationary

flame to the synchronously rotating rotor reference frame. The d-q axis currents

generation subsystem (D.4) generates d-q axis currents in the rotating reference frame

from the synchronously rotating d-q axis voltages and motor parameters. The actual

current generation subsystem was shown in Fig. 4.1, which generates actual currents ia,

ib and ic from the synchronously rotating d-q axis currents and the rotor position angle.

The motor output subsystem (D.5) generates the rotor position angle, th_e, and the

motor speed, r, from the synchronously rotating d-q axis currents and the motor

parameters using the motor dynamics.

66

D.1 Hysteresis Current Controller Subsystem

67

D.2 Inverter Subsystem

68

D.3 Coordinate Transformation Subsystem

69

D.4 d-q axis Current Generation Subsystem

70

D.5 Motor Output Subsystem

71

Appendix-E: Application of RK4 (ode4) to solve for id and iq from IPMSM Mathematical Model

% Name: Shahida Pervin% Matric #: SGP 110002% Supervisor: Dr. Zailan Siri% 4th order Runge-Kutta Method to solve for id and iq% Solve the 1st order differential Eqn. using ode4; % diq/dt=397.78-24.26iq-201.03id =f(t,iq,id);% did/dt=-3534.401-45.48id+706.83*iq = f(t,iq,id);% iq(n+1)=iq(n)+(k1+2k2+2k3+k4)h/6; iq(0)=5, t(0,1), h=0.0001, N=10000;% k1=hf(x0,t0); k2=h*f(iq0+k1/2, t0+h/2); k3=h*f(iq0+k2/2, t0+h/2);% k4=h*f(iq0+k3, t0+h);% t0=0;% iq0=10;

cleara=0;b=0.1;t=a;h=.0001;N=(b-a)/h;iq(1)=10;id(1)=-20;for i=1:Nk1(i)=h*(397.78-24.26*iq(i)-201.03*id(i));k2(i)=h*(397.78-24.26*(iq(i)+k1(i)/2)-201.03*id(i));k3(i)=h*(397.78-24.26*(iq(i)+k2(i)/2)-201.03*id(i));k4(i)=h*(397.78-24.26*(iq(i)+k3(i))-201.03*id(i));iq(i+1)=iq(i)+(k1(i)+2*k2(i)+2*k3(i)+k4(i))/6; k11(i)=h*(-3534.401-45.48*id(i)+706.83*iq(i));k12(i)=h*(-3534.401-45.48*(id(i)+k11(i)/2)+706.83*iq(i));k13(i)=h*(-3534.401-45.48*(id(i)+k12(i)/2)+706.83*iq(i));k14(i)=h*(-3534.401-45.48*(id(i)+k13(i))+706.83*id(i));id(i+1)=id(i)+(k1(i)+2*k2(i)+2*k3(i)+k4(i))/6;t=t+h;endt=a:h:b;plot(t,iq)hold onplot(t,id),grid

72